# Copyright (c) 2017 Weitian LI # MIT license """ Calculate the synchrotron emission and inverse Compton emission for simulated radio halos. References ---------- .. [cassano2005] Cassano & Brunetti 2005, MNRAS, 357, 1313 http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C Appendix.C .. [era2016] Condon & Ransom 2016 Essential Radio Astronomy https://science.nrao.edu/opportunities/courses/era/ Chapter.5 .. [you1998] You 1998 The Radiation Mechanisms in Astrophysics, 2nd Edition, Beijing Sec.4.2.3, p.187 """ import logging import numpy as np import scipy.special from scipy import integrate from scipy import interpolate from ...utils.units import (Units as AU, Constants as AC) logger = logging.getLogger(__name__) def _interp_sync_kernel(xmin=1e-3, xmax=10.0, xsample=256): """ Sample the synchrotron kernel function at the specified X positions and make an interpolation, to optimize the speed when invoked to calculate the synchrotron emissivity. WARNING ------- Do NOT simply bound the synchrotron kernel within the specified [xmin, xmax] range, since it decreases as a power law of index 1/3 at the left end, and decreases exponentially at the right end. Bounding it with interpolation will cause the synchrotron emissivity been *overestimated* on the higher frequencies. Parameters ---------- xmin, xmax : float, optional The lower and upper cuts for the kernel function. Default: [1e-3, 10.0] xsample : int, optional Number of samples within [xmin, xmax] used to do interpolation. Returns ------- F_interp : function The interpolated kernel function ``F(x)``. """ xx = np.logspace(np.log10(xmin), np.log10(xmax), num=xsample) Fxx = [xp * integrate.quad(lambda t: scipy.special.kv(5/3, t), a=xp, b=np.inf)[0] for xp in xx] F_interp = interpolate.interp1d(xx, Fxx, kind="quadratic", bounds_error=True, assume_sorted=True) return F_interp class SynchrotronEmission: """ Calculate the synchrotron emissivity from a given population of electrons. Parameters ---------- gamma : `~numpy.ndarray` The Lorentz factors of electrons. n_e : `~numpy.ndarray` Electron number density spectrum. Unit: [cm^-3] B : float The assumed uniform magnetic field within the cluster ICM. Unit: [uG] """ # The interpolated synchrotron kernel function ``F(x)`` within # the specified range. # NOTE: See the *WARNING* above. F_xmin = 1e-3 F_xmax = 10.0 F_xsample = 256 F_interp = _interp_sync_kernel(F_xmin, F_xmax, F_xsample) def __init__(self, gamma, n_e, B): self.gamma = np.asarray(gamma) self.n_e = np.asarray(n_e) self.B = B # [uG] @property def B_gauss(self): """ Magnetic field in unit of [G] (i.e., gauss) """ return self.B * 1e-6 # [uG] -> [G] @property def frequency_larmor(self): """ Electron Larmor frequency (a.k.a. gyro frequency): ν_L = e * B / (2*π * m0 * c) = e * B / (2*π * mec) => ν_L [MHz] = 2.8 * B [G] Unit: [MHz] """ nu_larmor = AC.e * self.B_gauss / (2*np.pi * AU.mec) # [Hz] return nu_larmor * 1e-6 # [Hz] -> [MHz] def frequency_crit(self, gamma, theta=np.pi/2): """ Synchrotron critical frequency. Critical frequency: ν_c = (3/2) * γ^2 * sin(θ) * ν_L Parameters ---------- gamma : `~numpy.ndarray` Electron Lorentz factors γ theta : `~numpy.ndarray`, optional The angles between the electron velocity and the magnetic field. Unit: [rad] Returns ------- nu_c : `~numpy.ndarray` Critical frequencies Unit: [MHz] """ nu_c = 1.5 * gamma**2 * np.sin(theta) * self.frequency_larmor return nu_c @classmethod def F(cls, x): """ Synchrotron kernel function. NOTE ---- * Use interpolation to optimize the speed, also avoid instabilities near the lower end (e.g., x < 1e-5). * Interpolation also helps vectorize this function for easier calling. * Cache the interpolation results, since this function will be called multiple times for each frequency. Parameters ---------- x : `~numpy.ndarray` Points where to calculate the kernel function values. NOTE: X values will be bounded, e.g., within [1e-5, 20] Returns ------- y : `~numpy.ndarray` Calculated kernel function values. References: Ref.[you1998] """ x = np.array(x, ndmin=1) y = np.zeros(x.shape) idx = (x >= cls.F_xmin) & (x <= cls.F_xmax) y[idx] = cls.F_interp(x[idx]) # Left end: power law of index 1/3 idx = (x < cls.F_xmin) A = cls.F_interp(cls.F_xmin) y[idx] = A * (x[idx] / cls.F_xmin)**(1/3) # Right end: exponentially decrease idx = (x > cls.F_xmax) y[idx] = (0.5*np.pi * x[idx])**0.5 * np.exp(-x[idx]) return y def emissivity(self, frequencies): """ Calculate the synchrotron emissivity (power emitted per volume and per frequency) at the requested frequency. NOTE ---- Since ``self.gamma`` and ``self.n_e`` are sampled on a logarithmic grid, we integrate over ``ln(gamma)`` instead of ``gamma`` directly: I = int_gmin^gmax f(g) d(g) = int_ln(gmin)^ln(gmax) f(g) g d(ln(g)) Parameters ---------- frequencies : float, or 1D `~numpy.ndarray` The frequencies where to calculate the synchrotron emissivity. Unit: [MHz] Returns ------- syncem : float, or 1D `~numpy.ndarray` The calculated synchrotron emissivity at each specified frequency. Unit: [erg/s/cm^3/Hz] """ j_coef = np.sqrt(3) * AC.e**3 * self.B_gauss / AU.mec2 # Ignore the zero angle theta = np.linspace(0, np.pi/2, num=len(self.gamma))[1:] theta_grid, gamma_grid = np.meshgrid(theta, self.gamma) nu_c = self.frequency_crit(gamma_grid, theta_grid) # 2D grid of ``n_e(gamma) * sin^2(theta)`` nsin2 = np.outer(self.n_e, np.sin(theta)**2) frequencies = np.array(frequencies, ndmin=1) syncem = np.zeros(shape=frequencies.shape) for i, freq in zip(range(len(frequencies)), frequencies): logger.debug("Calc synchrotron emissivity at %.2f [MHz]" % freq) kernel = self.F(freq / nu_c) # 2D samples over width to do the integration s2d = kernel * nsin2 # Integrate over ``theta`` (the last axis) s1d = integrate.simps(s2d, x=theta) # Integrate over energy ``gamma`` in logarithmic grid syncem[i] = j_coef * integrate.simps(s1d*self.gamma, np.log(self.gamma)) if len(syncem) == 1: return syncem[0] else: return syncem